deterministic size effect in concrete simulated with...

6th European Conference on Computational Mechanics (ECCM 6)7th European Conference on Computational Fluid Dynamics (ECFD 7)

11–15 June 2018, Glasgow, UK

DETERMINISTIC SIZE EFFECT IN CONCRETE SIMULATED WITHTWO VISCOPLASTIC MODELS

ANDRZEJ WINNICKI1, ADAM WOSATKO2 AND MICHAŁ SZCZECINA3

1 Faculty of Civil Engineering, Cracow University of TechnologyWarszawska 24, 31-155, Cracow, Poland

[email protected], www.wil.pk.edu.pl

2 Faculty of Civil Engineering, Cracow University of TechnologyWarszawska 24, 31-155, Cracow, Poland

[email protected], www.wil.pk.edu.pl

3 Faculty of Civil Engineering and Architecture, Kielce University of TechnologyTysiąclecia Państwa Polskiego Ave. 7, 25-314, Kielce, Poland

[email protected], www.wbia.tu.kielce.pl

Key words: Concrete, Size effect, Viscoplasticity, FEM

Abstract. The paper examines the ability of two selected viscoplastic models to reproduce thedeterministic size effect in plain concrete specimens. The first one is the concrete damagedplasticity model and it is available in the ABAQUS package. The second one is the Hoffmanviscoplastic consistency model and it is programmed in FEAP. Rate dependency existing in bothmodels serves as a localization limiter leading to mesh-objective results for reasonably highvalues of viscosity, so the reproducing of the deterministic size effect is expected. Numericalanalysis is performed for notched and unnotched beams under three point bending using bothmodels. Additionally, the results are compared with the experiment made by Grégoire et al.

1 INTRODUCTION

The origin of the knowledge concerning the size effect in brittle and quasi-brittle materialscan be traced back to the works of Galileo. Nowadays the subject has matured and usuallya distinction is made between deterministic and stochastic size effects. In brittle and quasi-brittle materials deterministic size effect is more important than the stochastic one originatingfrom the randomness of material strength [1]. There is extensive experimental evidence of thedeterministic size effect in concrete, e.g. [2, 3]. Since the development of fracture mechanicsthe knowledge of size effect laws has also matured [4].

The ability of the both non-local integral or gradient (either in plastic or damage format)models to recreate the size effect in numerical simulations for concrete is well known and docu-mented [5, 6]. Much less research has been done for viscoplastic models. Such models, similarto non-local or gradient models, can act as localization limiters and overcome in an effectiveway the spurious mesh dependency appearing in numerical solution for elements made of plain

Andrzej Winnicki, Adam Wosatko and Michał Szczecina

concrete [7]. However, knowledge concerning their ability to reproduce the deterministic sizeeffect is limited.

The aim of the paper is a numerical examination of the deterministic size effect in three-point bending for specimens made of plain concrete using two viscoplastic models. The firstone is the Concrete Damage Plasticity (CDP) model with optionally included viscous term. Thesecond one is the Hoffman viscoplastic consistency (HVP) model for concrete implemented inFEAP code [8] by authors. In the paper a new modified version of this model is presentedassuming a non-associated flow rule with the plastic potential of the Drucker-Prager type. Thenumerical results are compared with the experiments reported in [9]. Two types of concretebeam specimens are analyzed – notched and unnotched.

2 REVIEW OF APPLIED MODELS

2.1 Concrete damaged plasticity model

The first model which is applied in computations was originally proposed in [10] and nextenhanced in [11]. This model is distributed by ABAQUS software and called therein "damagedplasticity model for concrete and other quasi-brittle materials" [12], but here acronym CDP willbe used. The theory of the plastic flow is non-associated in the CDP model. An application ofviscous term and isotropic damage as additional components of the model is also possible.

The yield function is introduced in the stress space:

F pCDP =q + 3α p+ β(�̃p) 〈σmax〉 − γ 〈−σmax〉

1− α− σc(�̃pc ) = 0 (1)

where p = −13σii is the hydrostatic pressure and q =

√32sijsij is the Mises deviatoric measure

of stress tensor σ. The parameter α depends on the relation of the uniaxial compressive strengthf ′c to the biaxial compressive strength f

′bc. Function β(�̃

p) changes according to stress-strainrelationships defined separately for compression and for tension. The parameter γ decides aboutthe shape of the yield surface in the deviatoric plane. The subscript max refers to the maximumprincipal stress, so 〈σmax〉 = 12(σmax+ |σmax|). The example of the initial yield surface describedin the plane stress state is depicted in Fig. 1(a). The uniaxial compressive strength f ′c and theuniaxial tensile strength f ′t and relation f

′bc = 1.16f

′c are according to the numerical analysis

presented in this paper, see Table 2 and Fig. 4. Plastic potential in the CDP model flow is definedas follows:

GpCDP =[(e f ′t tanψ)

2 + q2]1/2

+ tanψ p (2)

where e is the eccentricity of the plastic potential flow and ψ is dilatancy angle. This functionis illustrated in Fig. 1(b) in the p− q plane for the exemplary data: e = 0.1 (that value is defaultin ABAQUS for CDP), f ′t = 3.88 MPa and ψ = 15

◦.In this paper the CDP model is equipped with a viscoplastic regularization according to the

Duvaut-Lions approach [13] for the viscoplastic strain rate �̇vp:

�̇vp = (�p − �vp) /µ (3)

2


(a) Yield surface F pCDP in initial stage. (b) Plastic potential GpCDP.

Figure 1: Exemplary yield and plastic potential functions for CDP model.

The viscous term for plastic strain �p activates if only the relaxation time parameter µ is largerthan zero. The theory of plasticity described above can be combined with isotropic damage, butthis coupling is not employed in the computations presented in the paper.

2.2 Hoffman viscoplastic consistency model

Usually the viscoplastic models follow the approaches proposed by Perzyna [14] or Duvaut-Lions [13] in which the viscoplastic strains are defined in explicit way using the viscosity pa-rameter (relaxation time). A quite different approach was proposed by Wang [15] and calledthe consistency viscoplasticity. In the consistency viscoplasticity the yield function can expandor shrink depending on the actual viscoplastic strain rate. The stress state is forced to remain onthe yield surface and the consistency condition is invoked. There is no need for an additionalequation defining a viscoplastic multiplier. Instead, in the consistency condition two separategeneralized material moduli appear: a classical plastic one h and a viscoplastic one s.

The Hoffman viscoplastic consistency (HVP) model for concrete uses the Burzyński-Hoffmanyield surface in its isotropic form which has been successfully employed in the analysis of con-crete structures [16]:

F vp = q2 − 3p(fc − ft)− fcft = 0 (4)

The yield surface (at the beginning of plastic processes) is presented in Fig. 2 (the plane stressstate and p− q plane) for the data given in Table 2.

It is assumed that two internal variables κc and κt exist which are both functions of theequivalent viscoplastic strain. They describe in a separate way hardening/softening behaviourin compression and tension, respectively. In addition, two more internal variables ηc and ηtdetermine the increase/decrease of compressive and tensile strengths due to the actual rate ofthe equivalent viscoplastic strain. Thus, the actual compressive and tensile strengths are:

fc = fc(κc, ηc) and ft = ft(κt, ηt) (5)

3


(a) Initial yield surface in plane stress state. (b) Initial yield surface on p− q plane.

Figure 2: Exemplary yield function F vp for HVP model.

The rates of the internal variables depend on the current stress and the rates of internal vari-ables κ and η:

κ̇c = gc(σ)κ̇ and κ̇t = gt(σ)κ̇ (6)

η̇c = gc(σ)η̇ and η̇t = gt(σ)η̇ (7)

In the above equations gc and gt are scalar functions of stress accounting for independent pro-cesses of damage in compression and tension.

In turn, κ̇ is defined as an equivalent viscoplastic strain (in the rate form) assuming workhardening, and in a similar way η depends on the first derivative of the viscoplastic strain (i.e.its rate is a function of the second derivative):

κ̇ = (σ : �̇vp) /q and η̇ = (σ : �̈vp) /q (8)

The dependence of fc on κc and ηc is formulated in a general way as:

fc = f′cHc(κc)Sc(ηc) (9)

where f ′c is the initial compressive strength. Similarly, the actual tensile strength is:

ft = f′t Ht(κt)St(ηt) (10)

Specific forms of these functions used in this paper are presented in Fig. 5.The strain rate is decomposed into its elastic and viscoplastic parts:

�̇ = �̇e + �̇vp (11)

and the generalized Hooke’s law is valid for the elastic part:

σ̇ =D : �̇e (12)

4


The viscoplastic flow is defined similarly to the classical rate independent plasticity using anotion of the plastic potential:

�̇vp = λ̇m where m =∂Gvp

∂σ(13)

In the original version of the consistency model for concrete [17] the associated flow rule wasassumed:

Gvp ≡ F vp (14)

There is an ample evidence that for concrete (at least for larger values of the hydrostatic pressurep) that the associated flow rule leads to excessive plastic dilatancy – much larger than encoun-tered in experiments [18]. As a remedy, the plastic potential in a form of the Prager-Druckersurface is proposed, where ψ is the dilatancy angle:

Gvp = q + tanψp (15)

At this stage of the development the examples presented in this paper, however, are computedusing the original version of the model in its associated form.

In order to establish the viscoplastic multiplier λ̇ the consistency equation is used, which inits final form reads:

Ḟ vp = n : σ̇ − hλ̇− sλ̈ = 0 (16)

where n = ∂F vp/∂σ, h is the classical generalized plastic modulus and s is the generalizedviscoplastic modulus. Due to the last term the consistency equation is no longer an algebraicequation for the viscoplastic multiplier, but a differential equation of the first order, to be solvedfor an appropriate initial condition. In the case when the functions Sc and St are constant, theirderivatives vanish and Eq. (16) reduces to the form known from the classical rate independentplasticity.

3 BEAM UNDER THREE POINT BENDING – NUMERICAL STUDY

3.1 Geometry and material model data

In the paper the size effect for viscoplastic models is verified for the beam test under threepoint bending. The geometry and material data as well as comparison with experimental resultsare taken from [9]. Two options are considered, i.e. the beam with and without the notch.The dimensions for all specimens are given in Table 1. Numerical simulations are performedfor a half of the domain due to the symmetry, see Fig. 3. The mesh density is consistent withthe sizes of the configuration, so the magnitude of finite elements changes proportionally tothe geometry of specimens. Plane stress is assumed. Thickness t = 50 mm is constant for allsimulations. Displacement control is employed in numerical analyses. Four-noded elementswith full integration are used as in the simulations in [9].

The material data for both models are listed in Table 2. In each model stress–strain relationsfor tension and compression are defined separately. The postcracking function for tension in theCDP model is determined as linear softening. It starts with strength f ′t = 3.88 MPa and goesto residual strength 0.01f ′t for ultimate strain �

crtu = 0.002. Stress–inelastic strain function for

5


Table 1: Beam test – geometry.

Specimen Length L Total height D Ligament height Dlig Span S Measurement base Lm[mm] [mm] [mm] [mm] [mm]

DN1 1400 400 320 1000 400DN2 700 200 160 500 200DN3 350 100 80 250 100DN4 175 50 40 125 50DU1 1400 400 1000 400DU2 700 200 500 200DU3 350 100 250 100DU4 175 50 125 50

(a) FE mesh for symmetric half of notchedbeam (N).

12Lm

(b) Symmetric half of unnotched beam (U)with FE mesh.

Figure 3: Beam test under three point bending.

Table 2: Beam test – material model parameters.

Young’s modulus: E = 37000 MPa Tensile strength: f ′t = 3.88 MPaPoisson’s ratio: ν = 0.21 Compressive strength: f ′c = 42.3 MPa

CONCRETE DAMAGED PLASTICITY (CDP) HOFFMAN VISCOPLASTICITY (HVP)Relation for compression: Fig. 4 Material function Hc: Fig. 5(a)Postcracking for tension: linear softening Material function Ht: Fig. 5(b)Ultimate tensile strain: �crtu = 2.0× 10−3Viscosity parameter: µ = 2.0× 10−4 s Material function Sc = St: Fig. 5(c)Dilatancy angle: ψ = 15◦ Work hardening Eq. (8)

compression is shown in Fig. 4. Dilatancy angle ψ equals 15◦. It is assumed that viscous termis activated in this model via parameter µ = 0.0002 s. The other model parameters are defaults,according to [12]. If the HVP model is taken into account, functions Hc(κc) for compressionand Ht(κt) for tension are also defined in different ways, cf. Figs 5(a) and 5(b). However,functions of the equivalent viscoplastic strain rate, i.e. Sc(ηc) for compression and St(ηt) fortension, remain the same in these two regimes, see Fig. 5(c).

6


0 0.001 0.002 0.0035

40

20

0

�inc

σc [MPa]

Figure 4: Stress–inelastic strain relation for compression applied in CDP model.

0

0.2

0.4

0.6

0.8

1

0 0.001 0.002 0.003 0.004

κc

Hc

(a) Function Hc.

0

0.2

0.4

0.6

0.8

1

0 0.0005 0.001 0.0015 0.002

κt

Ht

(b) Function Ht.

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2

ηc = ηt

Sc=St

(c) Function Sc = St.Figure 5: Material functions for Hoffman viscoplasticity.

3.2 Results for concrete damaged plasticity

The results for the CDP model are discussed at first. Fig. 6(a) shows of total force applied tothe beam with the notch versus crack mouth opening displacement (CMOD). They are depictedfor specimens DN1–DN4 and compared with experimental diagrams taken from [9]. It is seenthat the load carrying capacity obtained for the CDP model is quite close to the experimentalresult, however the character of post-peak equilibrium paths is different. Numerical resultsgive more ductile response and tending in the post-peak phase to a residual plateau rather thanceasing to zero. The CDP model exhibits a strong size effect reproducing behaviour of plainconcrete specimens in experiments in a good manner. In the diagrams presented in Fig. 6(b)so-called ligament stress is calculated in the following way:

σlig =3

2

F S

t (Dlig)2(17)

0

5

10

15

20

25

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

CMOD [mm]

Experiment [9]

DN1, 1400 x 400

DN2, 700 x 200

DN3, 350 x 100

DN4, 175 x 50Forc

e[k

N]

(a) Load–CMOD diagrams.

0

1

2

3

4

5

6

7

8

0 0.001 0.002 0.003 0.004 0.005 0.006

Horizontal strain

Lig

amen

tstr

ess

[MPa

]

DN4, 175 x 50

DN3, 350 x 100

DN2, 700 x 200

DN1, 1400 x 400

(b) Ligament stress–average strain diagrams.

Figure 6: Beam with notch – diagrams for CDP model.

7


(a) Specimen DN1. (b) Specimen DN2. (c) Specimen DN3. (d) Specimen DN4.

Figure 7: Beam with notch– distribution of equivalent tensile plastic strain (PEEQT) in final state for CDP model.

0

5

10

15

20

25

30

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Pseudo-CMOD [mm]

Forc

e[k

N]

Experiment [9]DU1, 1400 x 400DU2, 700 x 200DU3, 350 x 100

DU4, 175 x 50

(a) Load–pseudo-CMOD diagrams.

0

1

2

3

4

5

6

7

8

0 0.001 0.002 0.003 0.004 0.005 0.006

Horizontal strain

DU1, 1400 x 400

DU2, 700 x 200

Nom

inal

stre

ss[M

Pa]

DU4, 175 x 50

DU3, 350 x 100

(b) Nominal stress–average strain diagrams.

Figure 8: Unnotched beam – diagrams for CDP model.

(a) Specimen DU1. (b) Specimen DU2. (c) Specimen DU3. (d) Specimen DU4.

Figure 9: Unnotched beam – distribution of equivalent tensile plastic strain (PEEQT) in final state for CDP model.

where F is the force, S is the span, t is the thickness and Dlig is the ligament height in the notchregion of the beam. Horizontal strain is obtained from the CMOD divided by the base Lm.Fig. 7 presents contour plots of the distribution of the equivalent tensile plastic strain (PEEQT)for four beam sizes.

Analogical results, but for the unnotched beam are depicted in Figs 8 and 9. It is observedthat the load carrying capacity is overestimated in comparison to experiment [9], see Fig. 8(a).Here the so-called pseudo-CMOD is measured at the bottom surface between points Lm awayfrom each other. In Fig. 8(b) nominal stress is derived according to Eq. (17), but now the totalheightD is taken into account. Again, the size effect is noticed. Fig. 9 illustrates the distributionof the equivalent tensile plastic strain (PEEQT) for all specimens DU1–DU4.

8


0

1

2

3

4

5

6

7

8

0 0.001 0.002 0.003 0.004 0.005 0.006

Horizontal strain

Lig

amen

tstr

ess

[MPa

]DN4, 175 x 50

DN3, 350 x 100

DN2, 700 x 200

DN1, 1400 x 400

(a) Beam with notch.

0

1

2

3

4

5

6

7

8

0 0.001 0.002 0.003 0.004 0.005 0.006

Horizontal strain

Nom

inal

stre

ss[M

Pa]

DU4, 175 x 50

DU3, 350 x 100

DU2, 700 x 200

DU1, 1400 x 400

(b) Unnotched beam.

Figure 10: Ligament/nominal stress vs average strain diagrams for CDP model without active viscous term, µ = 0.

The diagrams presented in Fig. 10 are prepared for the CDP model without active viscousterm, i.e. the relaxation time parameter µ equals 0. It means that in these computations localversion of the model is employed and as expected the size effect does not occur.

3.3 Results for consistency viscoplasticity

Fig. 11 presents the diagrams for the HVP model in case of the beam with the notch. Com-puted post-peak equilibrium paths shown in Fig. 11(a) are much stiffer than the experimentalresults and very slowly tend to zero. The size effect is barely visible, see Fig. 11(b). Contourplots of the parameter κt which describes the equivalent plastic strain in tension are illustratedin Fig. 12. These crack patterns are noticeably different from those obtained for the CDP model(cf. Fig. 7) and seem to be not physically motivated. When viscous functions Sc = St givenin Fig. 5(c) are artificially increased 3.6 times the size effect becomes more evident (Fig. 13),however still remains much smaller than that from the experiment.

The results for unnotched beams are shown in Figs 14–15. The size effect manifests itselfonly at the peak and just after it all curves overlap, see Fig. 14(b). In this case contour plots ofthe internal variable κt in Fig. 15 also differ from the results computed for the CDP model.

When the viscous term is excluded by setting Sc = St ≡ 1.0 the size effect is absent asexpected, see the diagrams in Fig. 16. However, the larger maximum stress obtained for thebeam with the notch is to the contrary to the experimental evidence.

0

5

10

15

20

25

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

CMOD [mm]

Forc

e[k

N]

DN4, 175 x 50

DN3, 350 x 100DN2, 700 x 200

DN1, 1400 x 400Experiment [9]


0

1

2

3

4

5

6

7

8

0 0.001 0.002 0.003 0.004 0.005 0.006

Horizontal strain

Lig

amen

tstr

ess

[MPa

]

DN1, 1400 x 400

DN3, 350 x 100

DN2, 700 x 200

DN4, 175 x 50


Figure 11: Beam with notch – diagrams for HVP model.

9


(a) Specimen DN1. (b) Specimen DN2. (c) Specimen DN3. (d) Specimen DN4.

Figure 12: Notched beam – distribution of internal variable κt in final state for HVP model.

0

5

10

15

20

25

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

CMOD [mm]

Forc

e[k

N]

DN4, 175 x 50

DN3, 350 x 100DN2, 700 x 200

DN1, 1400 x 400Experiment [9]


0

1

2

3

4

5

6

7

8

9

0 0.001 0.002 0.003 0.004 0.005 0.006

Horizontal strain

Lig

amen

tstr

ess

[MPa

]

DN4, 175 x 50

DN3, 350 x 100

DN2, 700 x 200

DN1, 1400 x 400


Figure 13: Beam with notch – diagrams for HVP model with more active viscous term.

0

5

10

15

20

25

30

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Pseudo-CMOD [mm]

DU4, 175 x 50

DU3, 350 x 100

DU1, 1400 x 400

Experiment [9]

Forc

e[k

N]

DU2, 700 x 200

(a) Load–pseudo-CMOD diagrams.

0

1

2

3

4

5

6

7

8

0 0.001 0.002 0.003 0.004 0.005 0.006

Horizontal strain

Nom

inal

stre

ss[M

Pa]

DU4, 175 x 50

DU3, 350 x 100

DU2, 700 x 200

DU1, 1400 x 400

(b) Nominal stress–average strain diagrams.

Figure 14: Unnotched beam – diagrams for HVP model.

4 CONCLUSIONS

It is well-known that non-local constitutive models are capable of reproducing the deter-ministic size effect due to the presence of an internal length parameter. In concrete damagedplasticity (CDP) and Hoffman viscoplastic consistency (HVP) models it is done via activationof the viscous term. In the paper these viscoplastic models are confronted on the example of thebeam with and without the notch. Comparison of the size effect for both models and also ex-perimental data [9] is depicted in Fig. 17. Logarithmic scale is used for both axes. They behavein different ways. The CDP model exhibits a pronounced size effect not only for the reachedmaximum strength, but also in the post-peak regime, i.e. different values of the fracture energy

10


(a) Specimen DU1. (b) Specimen DU2. (c) Specimen DU3. (d) Specimen DU4.

Figure 15: Unnotched beam – distribution of internal variable κt in final state for HVP model.

0

1

2

3

4

5

6

7

8

0 0.001 0.002 0.003 0.004 0.005 0.006

Horizontal strain

Lig

amen

tstr

ess

[MPa

]

DN1, 1400 x 400

DN2, 700 x 200

DN3, 350 x 100

DN4, 175 x 50


0

1

2

3

4

5

6

7

8

0 0.001 0.002 0.003 0.004 0.005 0.006

Horizontal strain

Nom

inal

stre

ss[M

Pa]

DU2, 700 x 200

DU4, 175 x 50

DU3, 350 x 100

DU1, 1400 x 400

(b) Unnotched beam.

Figure 16: Ligament/nominal stress vs average strain diagrams for HVP model without active viscous term,Sc = St ≡ 1.0.

1

1.5

2

2.5

1 2 4 8

Experiment [9]

HVP HVP,10CDP

log [DN(i)/DN4]

log(σ

lig/f′ t)

i = 4 3 2 1


1

1.5

2

2.5

1 2 4 8

HVPCDP

log [DU(i)/DU4]

log(σ

nom/f′ t)

i = 4 3 2 1

Experiment [9]

(b) Unnotched beam.

Figure 17: Size effect plots.

as the parameter of ductility are noticeable. The response of the HVP model is not clear in thecontext of the size effect. In the beam test with notch the size effect for the initial adopted datais invisible. The ability of the model in this regard is demonstrated for functions of the equiv-alent viscoplastic strain rate Sc = St with a higher limit value, i.e. when the viscous term ismore active. Significant disagreement of softening zones between two considered viscoplasticmodels is noticed, so in the HVP model the non-associated flow rule should be introduced andfurther parametric studies are therefore necessary.

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[1] Z. P. Bažant and J. Planas. Fracture and Size Effect in Concrete and Other QuasibrittleMaterials. CRC Press, 1998.

[2] M. R. A. van Vliet. Size effect in tensile fracture of concrete and rock. Ph.D. dissertation,Delft University of Technology, Delft, 2000.

[3] J. G. M. van Mier. Concrete Fracture: A Multiscale Approach. CRC Press, Boca Raton,2013.

[4] Z. P. Bažant and Q. Yu. Universal size effect law and effect of crack depth on quasi-brittlestructure strength. ASCE J. Eng. Mech., 135(2):78–84, 2009.

[5] C. Le Bellégo, J. F. Dubé, G. Pijaudier-Cabot, and B. Gérard. Calibration of nonlocaldamage model from size effect tests. Eur. J. Mech. A/Solids, 22(1):33–46, 2003.

[6] R. de Borst and M. A. Gutiérrez. A unified framework for concrete damage and fracturemodels including size effects. Int. J. Fracture, 95(1-4):261–277, 1999.

[7] L. J. Sluys. Wave propagation, localization and dispersion in softening solids. Ph.D.dissertation, Delft University of Technology, Delft, 1992.

[8] R.L. Taylor. FEAP – A Finite Element Analysis Program, Version 7.4, User manual.University of California at Berkeley, Berkeley, 2001.

[9] D. Grégoire, L. B. Rojas-Solano, and G. Pijaudier-Cabot. Failure and size effect fornotched and unnotched concrete beams. Int. J. Num. Anal. Meth. Geomech., 37(10):1434–1452, 2013.

[10] J. Lubliner, J. Oliver, S. Oller, and E. Oñate. A plastic-damage model for concrete. Int. J.Solids Struct., 25(3):299–326, 1989.

[11] J. Lee and G.L. Fenves. Plastic-damage model for cyclic loading of concrete structures.ASCE J. Eng. Mech., 124(8):892–900, 1998.

[12] SIMULIA. Abaqus Manual (6.14). Dassault Systemes, Providence, RI, USA, 2014.[13] G. Duvaut and I. J. Lions. Les Inequations en Mechanique et en Physique. Dunod, Paris,

France, 1972.[14] P. Perzyna. Fundamental problems in viscoplasticity. In Recent Advances in Applied

Mechanics, volume 9, pages 243–377, New York, USA, 1966. Academic Press.[15] W. Wang. Stationary and propagative instabilities in metals - a computational point of

view. Ph.D., TU Delft, the Netherlands, 1997.[16] N. Bićanić, C. J. Pearce, and D. R. J. Owen. Failure predictions of concrete like materials

using softening Hoffman plasticity model. In H. Mang, N. Bićanić, and R. de Borst,editors, Computational modelling of concrete structures, Euro-C 1994, volume 1, pages185–198, Swansea, UK, 1994. Pineridge Press.

[17] A. Winnicki, C. J. Pearce, and N. Bićanić. Viscoplastic Hoffman consistency model forconcrete. Comput. & Struct., 79:7–19, 2001.

[18] P. A. Vermeer and R. de Borst. Non-associated plasticity for soils, concrete and rock.Heron, 29(3), 1984.

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